Limits at Infinity Bard College（限制在无穷远处巴德学院）.pdfVIP

Limits at Infinity Lecture Notes To understand the ideas in chapter 8, we will need to have a better understanding of limits at infinity. We begin with a few examples. #8% EXAMPLE 1 Find lim . 8Ä_ 8 SOLUTION Note that both the numerator and denominator of the fraction are approaching infinity. However, the ratio of two infinities may be anything—it depends on how they compare. Lets try plugging in some values for 8 : #8% 8 #8% 8 8 ! #% #Þ)) !! #!% ! #Þ!*) !!! #,!!% ,!! #Þ!!#! !,!!! #!,!!% !,!! #Þ!!!# For large values of 8 , the numerator #8% is almost exactly twice the denominator 8 , so the ratio approaches # . Whats happening is that the constants % and are becoming less and less important as 8 grows larger: two billion and four isnt very different from two billion. As 8 Ä _, these constants play no role in the outcome: #8% #8 lim œ lim œ #. è 8Ä_ 8 8Ä_ 8 In general, when finding a limit as 8 Ä _ , it is usually only necessary to pay attention to the largest term of a sum. For a polynomial, this is the term with the highest power of 8 : # 8 8 EXAMPLE 2 Find lim . $ 8Ä_ %8